You can say: "I have been invited to her house to have dinner." This means you were invited, but you have yet to go for dinner. "I was invited to her house for dinner." is suitable — but isn't clear as to whether or not you have attended the dinner as yet. "Have been" is probably a little clearer and more formal. If you have already been to the dinner, you would say, "I was invited to her house for dinner," or "I went to her house for dinner." I hope that is helpful! :)

I agree with Julia - use "I've been invited..." If you use "was", it sounds as if you were invited, then you decided not to go. The meaning isn't clear. "I am invited" sounds formal and factual. You can use this, but it sounds as if you don't expect a response from your mother. Reading the rest of your question, I think it'll help a lot if you look for present perfect exercises. This is a necessary part of English grammar.

Julia I have to ask from you?

I agree, too. The best form is the present perfect, 'I've been invited.' The present perfect is a really useful tense. Here we're using it to refer to something that has happened recently which has a direct bearing on the present moment.

I think it's "was"